Defining the Natural Logarithm with a Definite Integral

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we define the natural logarithm using a definite integral. We will accomplish this challenge by using straightforward ideas from logarithms, exponents, and derivatives.

Defining the Natural Logarithm with a Definite Integral

Having heard the terms ''log'' and ''natural log'', Fred wonders about trees and lumberjacks and river rafts … It doesn't take much to set Fred off on a tangent. Let's help him stay focused by defining the natural logarithm. The result we are after is:


definition_of_natural_logarithm


Don't worry, Fred. We will review logarithms in general and the number e on our way to this definition of the natural logarithm.

Logarithm Basics

Let's start this review of logs with a discussion of exponents. Like 102 means 10 times 10, which is 100. Here are some other powers of 10:

  • 103 = 1000
  • 101 = 10

We can also have negative powers of 10. For example:

  • 10-2 = 1/(102) = 1/100 = .01
  • 10-1 = 1/10 = .1

And 100 = 1.

How does this relate to logarithms? We can read logarithms with a question. Fred is intrigued because he is always asking questions. When we read log10 100 the question is ''10 to what power gives 100?''. The answer to the question: 10 to the power 2 gives 100. Thus, log10 100 = 2. Some other log10 to interpret:

  • log10 1000 = 3
  • log10 10 = 1
  • log10 .01 = -2

And log10 1 = 0.

Something interesting happens when we write log10 103. The answer, 3, is the exponent. In general, we could write log10 10x = x.

Logarithm and exponentiation are inverse operations: they undo each other.

The base in the statement log10 is the number 10. Other numbers are also used as the base. The natural number, e, is hugely important as a base for logarithms. Instead of writing loge we write ln.

The inverses are now ln ex = x and eln x = x.

Fred would like a bit more review on the natural number, e. Some factoids:

  • e is an irrational number
  • e = 2.718281…
  • the derivative of ex is ex
  • the derivative of eax is eax times the derivative of ax

Deriving the Natural Logarithm Expression

Recall how e and ln are inverse operations. One undoes the other. We start with:


inverse_ln_and_e


Now, take the derivative of both sides:


derivative_of_both_sides


Remember how the derivative of eax is eax times the derivative of ax? Replace ax with ln x. On the left-hand side, the derivative of eln x is eln x times the derivative of ln x:


derivative_of_LHS


The right-hand side is the derivative of x which is 1.


derivative_of_RHS


So far, we have something times a quantity in a bracket is equal to 1. The something is eln x which is x. Thus,


x_bracket_equals_1


Dividing both sides by x:


the_antiderivative_expression


If Fred reads this statement, he notes the derivative of ln x is 1/x. Furthermore, the anti-derivative is what you differentiate to get the function. So the anti-derivative of 1/x is ln x.

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